Almost all k-colorable graphs are easy to color
Journal of Algorithms
The solution of some random NP-hard problems in polynomial expected time
Journal of Algorithms
A spectral technique for coloring random 3-colorable graphs (preliminary version)
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
Coloring random and semi-random k-colorable graphs
Journal of Algorithms
Algorithmic theory of random graphs
Random Structures & Algorithms - Special issue: average-case analysis of algorithms
Algorithms for coloring semi-random graphs
Random Structures & Algorithms
Hiding cliques for cryptographic security
Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms
A Generalized Encryption Scheme Based on Random Graphs
WG '91 Proceedings of the 17th International Workshop
Minimum coloring random and semi-random graphs in polynomial expected time
FOCS '95 Proceedings of the 36th Annual Symposium on Foundations of Computer Science
Heuristics for Finding Large Independent Sets, with Applications to Coloring Semi-random Graphs
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Algorithms for Colouring Random k-colourable Graphs
Combinatorics, Probability and Computing
Approximate graph coloring by semidefinite programming
SFCS '94 Proceedings of the 35th Annual Symposium on Foundations of Computer Science
Hi-index | 0.01 |
We present a simple BFS tree approach to solve partitioning problems on random graphs. In this paper, we use this approach to study the k-coloring problem. Consider random k-colorable graphs drawn by choosing each allowed edge independently with probability p after arbitrarily partitioning the vertex set into k color classes of "roughly equal" (i.e. Ω(n)) sizes. Given a graph G and two vertices x and y, compute n(G,x,y) as follows: Grow the BFS tree (in the subgraph induced by V - y) from x till the l-th level (for some suitable l) and find the number of neighbors of y in this level. We show that these quantities computed for all pairs are sufficient to separate the largest or smallest color class. Repeating this procedure k-1 times, one obtains a k-coloring of G with high probability, if p ≥ n-1+Ɛ, Ɛ ≥ X/√log n for some large constant X. We also show how to use this approach so that one gets even smaller failure probability at the cost of running time. Based on this, we present polynomial average time (p.av.t. ) k-coloring algorithms for the stated range of p. This improves significantly previous results on p.av.t. coloring [13] where Ɛ is required to be above 1/4. Previous works on coloring random graphs have been mostly concerned with almost surely succeeding (a.s.) algorithms and little work has been done on p.av.t. algorithms. An advantage of the BFS approach is that it is conceptually very simple and combinatorial in nature. This approach is applicable to other partitioning problems also.